MLDissertation.hpp File Reference

Plans regarding the PhD-dissertation of ML. More...

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Detailed Description

Plans regarding the PhD-dissertation of ML.

Improving notations
  • Tuples are to be written using round brackets; so likely all angle-brackets are to be replaced (they are only used as operators, for example constructors of partial assignments).
  • Proofs need to use to mark the end.
Improving usage of Latex
  • All sections etc. need labels.
  • And also all definitions, lemmas etc. need labels.
  • There are missing citations.
  • There are undefined references.
Section "Category of literal structures"
  • This section is the first to be completed.
  • Along the way, the definitions needed for categorical and algebraic structures are to be provided/completed.
  • Literature:
    1. Likely, "literal structures" exist already in the literature --- under what names?
    2. And is the category somewhere studied?
  • Further properties (to complete the picture):
    1. How many literal structures exist on a set?
    2. For a finite set, how can we enumerate them?
    3. How many isomorphism types exist on a set?
    4. How can we enumerate them?
    5. Can we efficiently decide the isomorphism problem?
    6. More generally, can we count homomorphism?
    7. Can we enumerate them?
    8. Is the homomorphism-existence-problem efficiently decidable?
Correction of errors
  • During our last meeting, we discussed a false proof --- has this been corrected?
  • Lemma {lem:MBCPNeverTough}:
    1. First precise definitions of "bicliques", "maximal bicliques" and "biclique partitions" are needed; compare with the definitions in the OKlibrary.
    2. One auxiliary lemma: If a biclique partition contains a star, then without increasing the number of bicliques one can make the star maximal.
    3. Make the argumentation of the lemma more precise, using the above definitions and the auxiliary lemma.
Transfer to part "Dissertation"
  • It is of utmost importance that finalised definitions, lemmas etc. are transferred to Part III, where they are then carefully checked.
  • This must start immediately.

Definition in file MLDissertation.hpp.